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Morse and semistable stratifications of Kähler spacesby \({\Bbb C}^{\ast}\)-actions

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Abstract.

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \({\Bbb C}^{\ast}\)-Kähler manifold coincide if the moment map is proper and if the fixed points set \(X^{{\Bbb C}^{\ast}}\) has a finite number of connected components. For general Kähler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.

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Authors and Affiliations

  1. Tsinghua University, Beijing, P.R. China

    Qilin Yang

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  1. Qilin Yang
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Author’s address: Department of Mathematics, Tsinghua University, 100084 Beijing, P.R. China

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Yang, Q. Morse and semistable stratifications of Kähler spacesby \({\Bbb C}^{\ast}\)-actions. Monatsh Math 155, 79–95 (2008). https://doi.org/10.1007/s00605-008-0543-3

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  • DOI: https://doi.org/10.1007/s00605-008-0543-3

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